Integral factorial ratios

Abstract

This paper is concerned with the problem of classifying tuples of natural numbers $a_1,\dots ,a_K$ and $b_1,\dots ,b_L$ such that the ratio of factorials ${\prod }_{i=1}^K(a_{i}n)!∕{\prod }_{j=1}^L(b_{j}n)!$ is an integer for all natural numbers n. A complete solution to this problem is only known in the case $L-K=1$, due to the work of Bober building on an observation of Rodriguez Villegas, which relies on the Beukers–Heckman classification of algebraic hypergeometric functions. We provide an alternative proof of this result, which also makes progress on the general problem when $L-K>1$.